1. Betatron Maria Kazachenko Physics department Montana State University

2. Any sufficiently advanced technology is indistinguishable from magic.
What is betatron?
Any sufficiently advanced technology is indistinguishable from magic.
Arthur C. Clarke

3. Introduction Donald Kerst; e- accelerator; 1940 New
Particle accelerator that uses the electric field induced by a varying magnetic field to accelerate electrons to high speeds in a circular orbit.
Used in
Nuclear reactions
X-ray sources in medicine
Possible solar flare mechanism
New
e- acceleration with EM induction
CR source
Energy
Supernova
1014 eV
Sun
105 eV
Milky Way
108 eV
Betatron
Before: fast e- - only in cosmic rays

4. Outline Methods of electrons acceleration (historically)
Van de Graaf high voltage generator (E=const, B=const)
too big
2. Linear accelerator (E changes, B=const)
too long
3. Circular accelerator (E changes, B=const)
relativistic effects
4. Betatron accelerator (B changes, vortex E)
How it works?
Magnetic field distribution
Equilibrium orbit and stability
Electron injection
5. Conclusion

5. Use multiple acceleration with lower potential?
Before a betatron
Why do we need to accelerate particles?
To measure smth small requires smth smaller
De Broglie and wave-particle dualism
Particle acceleration in electric field
Is it possible to get
5 MeV KE without using
5 MV potential?
Use multiple acceleration with lower potential?
Nature:
Beta-radioactive materials;
Human:
vacuum tube
electron gun
Van de Graaf generator
Disadvantage: single acceleration, size

6. Linear Accelerator 0 V +1000 V +2000 V +3000 V -1000 V +1000 V -1000 V
KE=3000 eV e
-1000 V
+1000 V e
-1000 V
+1000 V
-1000 V
+1000 V e e
-1000 V
+1000 V
To get KE=106eV, we need 1000 V not 106V.
If 1000 plates, KE=1000*Vsingle_pair=106eV

7. Sloan and Kots got mercury ions accelerated up to 2.85 MeV;
Linear Accelerator
X-rays e e
High voltage
ion source
Accelerating
plates
Vacuum
chamber
Source of radio
frequency (RF)
Target
Sloan and Kots got mercury ions accelerated up to 2.85 MeV;
1.85 meter linac
36 electrodes
Could be ~1 km, easily!

8. An Early Circular Accelerator
In 1929, Ernest Lawrence developed the first circular accelerator
This cyclotron was only 4 inches in diameter, and contained two D-shaped magnets separated by a small gap
An oscillating voltage created an electric field across the small gap, which accelerated the particles as they went around the accelerator

9. Why can’t we use cyclotron to accelerate electrons?
Time period
Proton MeV
Electron 25 KeV
Impossible to accelerate electrons in cyclotron up to several million of eV

10. E- acceleration with EM induction
e- rotating in a circle in magnetic field B
After one revolution Ekin increases by
t=0.001 seconds, S=290 km, 18.5 MeV, revolutions
- How can we make e- rotate in a circle?
- Using special configuration of magnetic field.
; if

11. Basic principle of how the betatron works
Conclusion: Electron will have circular motion of constant radius if the half of the average of the magnetic field within the circle is equal to the value of magnetic field on the orbit.
Special B (r) distribution
Time evolution of the magnetic field

12. Stability of motion on the equilibrium orbit
Is motion on the equilibrium orbit stable? S=300 kilometers!!! T=1/1000 sec
1. Radial stability
stable
unstable
2. Axial stability
Barrel-type magnetic field lines
Lorentz force deflects electrons back to the median plane.

13. First betatron. Electron injection.
Ausserordentlichhochgeschwindigkeitelektronenentwickelndenschwerarbeitsbeigollitron
German for "extraordinarily high-speed electron generator".
How to realize the initial condition in practice?
B=B(t) => very short time
when B~B0

14. Summary Betatron in use (in the past)
Instrument
Shape
Electric field
Magnetic field
Electron energy,
MeV
Van de Graaf generator
linear
constant 25
Linear accelerator
variable
2.85
(50.000)
Cyclotron
circle
0.025
Betatron
torus
300
Synchrotron
10.000
Betatron in use (in the past)
Fast electrons in particle physics
X-rays (radiation oncology)
Best e--accelerators now
Large electron-positron collider – 8*104 MeV
International Linear Collider, 106 MeV

15. Questions?

16. Syncrotron radiation

17. Magnetic mirror
A magnetic mirror is a magnetic field configuration where the field strength changes when moving along a field line.

18. Adiabatic invariants Particle Trapping
For periodic motion, the adiabatic invariants are the action integrals taken over period of the motion.
First adiabatic invariant
Magnetic moment cons-n
in time-dependent B
(cyclotron motion)
Second adiabatic invariant
(longitudinal motion)
Particle Trapping

19. Magnetic mirror:
magnetic field configuration where the field strength changes when moving along a field line, as a result charged particles bounce back from the high field region.
Fermi acceleration:
Decrease of the field line length provides the first-order Fermi acceleration
Betatron acceleration
Compression of the magnetic field lines provides betatron acceleration

20. Particle Acceleration in a Collapsing Trap
A magnetic trap between the Super-Hot Turbulent-Current Layer (SHTCL) and a Fast Oblique Colisionless Shock (FOCS) above magnetic obstacle (MO)
Particles are captured into a collapsing magnetic trap where they accelerate further to high energies.
Apart from the First-Order Fermi acceleration the authors have suggested taking into account the betatron effect in collapsing traps, i.e. an increase in the transverse momentum as the trap contracts.
Main idea of the paper:
to develop a trap model in which both Fermi and betatron accelerations are at work, compare efficiencies, pitch-angle distributions, total kinetic energy of trapped electrons.
Ref.: Somov, B.V. and Kosugi, T., ApJ, 485, 859, 1997

21. The formation of a trap. Its contraction. Particle acceleration
Electron energy in the magnetic reconnection region (RR) increases from a coronal thermal energy of 0.1 keV at least to an energy of 10keV.
Each magnetic flux tube is a trap since Bm>B0.
Particle injection is impulsive, i.e. electrons fall into trap at the initial time and subsequently either precipitate into the loss cone or become trapped, acquiring additional energy.
Due to motion from RR to chromosphere, the length of the trap decreases => particles energy in a trap increases
due to Fermi mechanism. When magnetic trap contracts transversely, particles are accelerated by betatron mechanism.

22. Transverse contraction changes from at which
to at which b(t)=bm
The change in the trap length l with time changes from l(0)=1 to l=0 or to some residual trap length.
Longitudinal invariant:
Transverse invariant:
As a result:
When two mechanisms act, the pitch angle is:
The nonrelativistic KE:
Pitch angle when particle falls into loss cone:
Kinetic energy at the escape time:

23. Gyrosynchrotron Radiation